Given: Sample space is the set of first 100 natural numbers.
n (S) = 100
Let ‘A’ be the event of choosing the number such that it is divisible by 4
n (A) = [100/4]
= [25]
= 25 {where [.] represents Greatest integer function}
P (A) = n (A) / n (S)
= 25/100
= 1/4
Let ‘B’ be the event of choosing the number such that it is divisible by 6
n (B) = [100/6]
= [16.67]
= 16 {where [.] represents Greatest integer function}
P (B) = n (B) / n (S)
= 16/100
= 4 /25
Now, we need to find the P (such that number chosen is divisible by 4 or 6)
P (A or B) = P (A ∪ B)
By using the definition of P (E or F) under axiomatic approach (also called addition theorem) we know that:
P (E ∪ F) = P (E) + P (F) – P (E ∩ F)
∴ P (A ∪ B) = P (A) + P (B) – P (A ∩ B)
[Since, we don’t have the value of P (A ∩ B) which represents event of choosing a number such that it is divisible by both 4 and 6 or we can say that it is divisible by 12.]
n (A ∩ B) = [100/12]
= [8.33]
= 8
P (A ∩ B) = n (A ∩ B) / n (S)
= 8/100
= 2/25
∴ P (A ∪ B) = P (A) + P (B) – P (A ∩ B)
P (A ∪ B) = 1/4 + 4/25 – 2/25
= 1/4 + 2/25
= 33/100
